Finite Group Databasebeta

← #4244729

Subgroups — #4244729

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17243C211yesmaximal, characteristic, complemented
2#17044C212yesmaximal, complemented
3#16953C212yesmaximal
4#16759C212yesmaximal
5#16096C211yesmaximal, characteristic, complemented
6#16093C212yesmaximal, complemented
7#16002C211yesmaximal, characteristic, complemented
8#16001C212yesmaximal, complemented
9#15981C211yesmaximal, characteristic, complemented
10#15904C212yesmaximal
11#15884C212yesmaximal, complemented
12#15728C212yesmaximal
13#15670C212yesmaximal
14#15178C211yesmaximal, characteristic
15#15063C211yesmaximal, characteristic
16#15017C211yesmaximal, characteristic
17#14604C212yesmaximal, complemented
18#14378C212yesmaximal, complemented
19#14362C212yesmaximal
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C2×#400C2212yes—
23C2×#397C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#387C2212yes—
28C2×#387C2212yes—
29C2×#387C2212yes—
30C2×#382C2211yescharacteristic
31C2×#382C2212yes—
32C2×#380C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1446C2212yes—
35#1446C2212yes—
36#1446C2212yes—
37#1445C2212yes—
38#1445C2211yescharacteristic
39#1445C2211yescharacteristic
40#1444C2212yes—
41#1444C2211yescharacteristic
42#1444C2211yescharacteristic
43#1444C2212yes—
44#1444C2211yescharacteristic
45#1444C2211yescharacteristic
46#1443C2212yes—
47#1443C2212yes—
48#1443C2212yes—
49#1443C2212yes—
50#1443C2212yes—
51#1443C2212yes—
52#1442C2212yes—
53#1442C2212yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2212yes—
59#1442C2212yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2211yescharacteristic
64#1442C2212yes—
65#1442C2211yescharacteristic
66#1441C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2212yes—
73#1440C2212yes—
74#1440C2212yes—
75#1440C2212yes—
76#1439C2212yes—
77#1439C2212yes—
78#1439C2211yescharacteristic
79#1439C2212yes—
80#1439C2212yes—
81#1439C2211yescharacteristic
82#1439C2211yescharacteristic
83#1439C2211yescharacteristic
84#1439C2212yes—
85#1437C2212yes—
86#1437C2212yes—
87#1437C2212yes—
88#1436C2212yes—
89#1436C2212yes—
90#1436C2212yes—
91#1436C2212yes—
92#1436C2212yes—
93#1436C2212yes—
94#1436C2212yes—
95#1436C2212yes—
96#1436C2212yes—
97#1436C2212yes—
98#1436C2212yes—
99C4⋊#95C2211yescharacteristic
100#1434C2211yescharacteristic
101#1434C2212yes—
102#1434C2212yes—
103#1434C2212yes—
104#1434C2212yes—
105C22⋊#95C2212yes—
106C22⋊#95C2211yescharacteristic
107C23×Q8C2312yes—
108C24×C4C2311yescharacteristic
109C2×C4⋊Q8—42no—
110C2×C4⋊Q8—44no—
111C2×C22⋊Q8—28no—
112C2×C22⋊Q8—44no—
113C2×C22⋊Q8—44no—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2312yes—
142#400—28no—
143#397—44no—
144#391—28no—
145#391—44no—
146#389—44no—
147#387—44no—
148#387—44no—
149#387—44no—
150#382—42no—
151#382—44no—
152#380—42no—
153C4×C22⋊C4—42no—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C2×#95C2312yes—
207C2×#95C2312yes—
208C2×#95C2312yes—
209C2×#95C2311yescharacteristic
210C2×#95C2311yescharacteristic
211C25C2411yescharacteristic
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8—42no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—42no—
223C23×C4—41no—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C4⋊Q8—416no—
243C22⋊Q8—432no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4#4118yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—42no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—22no—
341#95—24no—
342#95—22no—
343#95—42no—
344#95—42no—
345#95—42no—
346#95—42no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—42no—
354#95—41no—
355#95—42no—
356#95—42no—
357#95—42no—
358#95—42no—
359#95—42no—
360#95—42no—
361#95—41no—
362#95—41no—
363#95—42no—
364#95—42no—
365#95—42no—
366#95—42no—
367#95—42no—
368#95—42no—
369#95—42no—
370#95—42no—
371#95—42no—
372#95—42no—
373#95—42no—
374#95—42no—
375#95—42no—
376#95—42no—
377#95—42no—
378C24—22no—
379C24—22no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—21no—
388C242−1+412yes—
389C24C2511yescharacteristic
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—84no—
395C2×Q8—84no—
396C2×Q8—84no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—48no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—84no—
506C22×C4—82no—
507C22×C4C2×#4114yes—
508C22×C4C2×#4114yes—
509C22×C4C2×#4114yes—
510C22×C4H(5,𝔽2)14yes—
511C22×C4H(5,𝔽2)12yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C42−1+412yes—
514C22×C42−1+412yes—
515C22×C42−1+412yes—
516C4⋊C4—216no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—84no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—44no—
534C22⋊C4—48no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C42—44no—
540C42—48no—
541C42—44no—
542C42—84no—
543C42—84no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—22no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23C22×#4112yes—
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×H(5,𝔽2)12yes—
571C23C2×2−1+411yescharacteristic
572C23C2×2−1+411yescharacteristic
573C23C2×2−1+411yescharacteristic
574C23C2×2−1+411yescharacteristic
575C23#58511yescharacteristic
576C23#58512yes—
577C23#58512yes—
578Q8—816no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—88no—
693C2×C4#55914yes—
694C22—44no—
695C22—42no—
696C22—42no—
697C22—44no—
698C22—42no—
699C22—81no—
700C22—82no—
701C22—82no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22C2×#55911yescharacteristic
706C22#349012yes—
707C22#349412yes—
708C22#351512yes—
709C22#351512yes—
710C22#351612yes—
711C22#351612yes—
712C22#353211yescharacteristic
713C22#353411yescharacteristic
714C22#353811yescharacteristic
715C22#353811yescharacteristic
716C22#354012yes—
717C22#354112yes—
718C22#354112yes—
719C22#354712yes—
720C22#355011yescharacteristic
721C22#355111yescharacteristic
722C22#355111yescharacteristic
723C22#355211yescharacteristic
724C22#355511yescharacteristic
725C22#356112yes—
726C22#356311yescharacteristic
727C22#356612yes—
728C4—24no—
729C4—42no—
730C4—44no—
731C4—44no—
732C4—42no—
733C4—42no—
734C4—44no—
735C4—42no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—81no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C2—82no—
747C2#4317511yescharacteristic
748C2#4522611yescharacteristic
749C2#4522611yescharacteristic
750C2#4742212yes—
751C2#5010212yes—
752C2#5023712yes—
753C2#5125712yes—
754C2#5321911yescharacteristic
755C2#5655811yescharacteristic
756C2#5826111yescharacteristic
757C2#5888911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17243C2semidirect
2. #17044C2semidirect
5. #16096C2semidirect
6. #16093C2semidirect
7. #16002C2semidirect
8. #16001C2semidirect
9. #15981C2semidirect
11. #15884C2semidirect
17. #14604C2semidirect
18. #14378C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#172432
#170442
#169532
#167592
#160962
#160932
#160022
#160012
#159812
#159042
#158842
#157282
#156702
#151782
#150632
#150172
#146042
#143782
#143622